Point group

A point group consists of geometric transformations known as symmetry operations, which preserve a single common point while transforming an object defined in a real vector space into physically indistinguishable replicas of itself.

Although an object undergoing a symmetry operation ends up looking the same after the transformation, the labelling of similar components of the object may change. In other words, to form a point group, all symmetry operations for an object must:

    1. Send the object into physically indistinguishable copies of itself.
    2. Combine with one another through binary operations such that the results are consistent with the 4 properties of a group.
    3. Leave one point invariant.

Point groups are determined by considering symmetry operations for different objects, beginning with simple shapes and moving on to more complex ones. According to the three abovementioned requirements, we start by inspecting the chosen object visually and finding all the symmetry elements (not to be confused with group elements) and their associated symmetry operations. For example, the only symmetry elements for the object in figure I, which is made up of two equally spaced right-angled triangles on a circle, are the identity symmetry element  and a 2-fold rotation axis . The corresponding symmetry operations are  and .

Next, we select a position vector  (see figure II) and perform the symmetry operations  and consecutively on the position vector. The results, in relation to the transformation of the position vector, are summarised in the multiplication table i:

Note that we have omitted the carets – i.e. – for simplicity. From the table, we can easily verify all four properties of the group, e.g. the identity element is and the inverse of an element of the group is the element itself. Therefore, the set of symmetry operations for the object forms a point group of order 2 under the binary operation of multiplication. We call this group the  point group. Similarly, the set of symmetry operations  for the object in figure III forms the point group  of order 3 (see multiplication table ii). In general, we have an infinite number of uniaxial point groups , each of which is called a cyclic group, whose elements are . For a cyclic group where is even, one of its elements is equivalent to the symmetry operation .

Let’s suppose the object in figure I have complete arrow heads (see figure IV). Other than  and , the object has a plane of symmetry perpendicular to the axis of rotation (the horizontal plane is denoted by ) and a centre of inversion at the origin. The set of symmetry operations  forms the Abelian point group  (see multiplication table iii). The object in figure IVa belongs to the  point group (see multiplication table iv) with the set of elements , where . Similar to the point groups, if we apply the same logic to other related objects, we have an infinite number of point groups, each with symmetry operations of if  is even (one of the  symmetry operations is equivalent to ) and if is odd.


Why aren’t and elements of the group ?


They are not unique elements, as they are equivalent to other ‘simpler’ elements of the group:


For an object that is made up of two equally spaced equilateral (or isosceles) triangles on a circle (see figure V above), we have the Abelian point group , whose elements are the symmetry operations , ,  and , where the symmetry elements for  and  are the vertical planes: -plane and -plane respectively. The multiplication table for this point group is shown in table v above. As before, we have an infinite number of  point groups. The cone depicted in figure VI is an example of an object that belongs to the  point group, where , i.e. the  point group, with the set of symmetry operations: .


Are and point groups?


is a trivial point group whose sole element is the symmetry operation . An object of this group is considered to have no rotational symmetry.

and  have the same set of symmetry operations: . Since objects of both point groups have no rotational symmetry, the symbol for the reflection symmetry operation does not have a subscript. In fact, these point groups are so unique that they are collectively known as the point group (for Spiegel, the German word for mirror).

Another unique point group not mentioned above is , whose elements are the symmetry operations and .  and  are known as the non-axial point groups.


The next few related point groups are and (for dihedral). They are related in the sense that they have in common the elements  and , which serve as identifiers in categorising molecules by point groups.



Object Group elements


The object is cyclohexane in its twisted boat form. A rhombic disphenoid (tetrahedron with scalene triangles as its faces) also belong to this point group.


The object is a tetragonal disphenoid (faces are isosceles triangles). is a dihedral plane, i.e. a vertical plane that bisects the angle between two  axes.
1)   axis to screen and bisects .
2)    axes, i) bisects  and ; ii) bisects  and .
3)   planes to screen, i) through ; ii) though .

The  and point groups are identical to the  and point groups respectively in non-standard orientation, i.e. the principal axis is along the -axis. A dumbbell has symmetry elements that are associated with the symmetry operations of

which is a special point group like .

Next, we have the  point group, which is in general associated with the symmetry operations . When , we have the point group , which is the same as the point group . When , the point group  is identical to the point group . For , we need to analyse the point group  with odd and even  separately.

Consider an  point group . When  is odd, the symmetry operation . If , then we have or (according to the closure property of a group). Since , then . Moreover, , which implies that . We can rewrite the symmetry operations of the  point group ( is odd) as

which is equivalent to the set of elements of the point group . For example, the object in figure IVa belongs to the point group and hence to the point group .

When is even, the symmetry operation . Since , we have or . Similarly, or . This implies that . The elements of can be denoted by , where can be odd or even. If is even, then . For example,  and . If is odd, then . We can therefore express the symmetry operations for the  point group (when  is even) as . Figure VII depicts an object that belongs to the  point group.

Taking into account the above characteristics of the  point group, it is possible to relabel it as the  point group, where .

The rest of the point groups are the tetrahedral groups , the octahedral groups , the icosahedral groups  and the special orthogonal group in 3-dimensions  (also known as the full rotation group). The tetrahedral and octahedral groups are collectively called the cubic groups.

Symbol Object Group elements Notes
Each of the three  axes passes perpendicularly through the centre of one of the three depicted faces, e.g. . Each of the four  axes passes through one of four body diagonals, e.g. .

or simply

Same rotation axes as . Three  axes (each with 2 symmetry operations: ) coincident with the  axes. Each of the six  passes through two diagonally opposite edges of the cube.
Same rotation axes as . Four axes (each with 2 symmetry operations: ) coincident with the  axes. A centre of symmetry  and three : i) bisecting  and , ii) bisecting  and , iii) bisecting  and .

or simply

Same as , but the  axes are now  axes (each with 3 symmetry operations: ). Six  axes through mid-points of diagonal edges, e.g.  and .
Same as , but includes centre of inversion ,  axes of  and mirror planes of  and .
The object is a snub dodecahedron with 92 faces (12 pentagons, 80 equilateral triangles), 150 edges and 60 vertices.

The object is a truncated icosahedron with 32 faces (12 pentagons, 20 hexagons), 90 edges and 60 vertices. Same symmetry elements as  with the addition of a centre of inversion, improper axes and mirror planes.

, all possible rotations The object is a sphere with an infinite number of rotation axes, each with all possible values of .



Why are  and  called tetrahedral point group and octahedral point group respectively?


A regular tetrahedron and a regular octahedron have all the same symmetry elements as those used to derive the  point group and the  point group respectively. A tetrahedron with reduced symmetry (e.g. with figure IVa attached to its faces) belongs to the  point group. Similarly, an octahedron with attachments to its vertices belong to the  point group.



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